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A Dynamical System Describing Evolution of the Implicit Surfaces in Incompressible Viscous Liquids

In: Numerical Mathematics and Advanced Applications

Author

Listed:
  • Petr Klouček

    (Rice University, Department of Computational and Applied Mathematics)

  • Michel V. Romerio

    (École Polytechnique Fédérale de Lausanne, Institut d’Analyse et Calcul Scientifique)

  • Jennifer L. Wightman

    (Rice University, Department of Computational and Applied Mathematics)

Abstract

Summary We present an equation capable of describing the evolution of implicit surfaces (boundaries of gas bubbles) in liquids. The equation does not contain any convective term and thus it is independent of the velocity and pressure fields, given by e.g. Navier-Stokes equations. The equation itself is obtained by a series of variational problems. First, by interpreting the Euler implicit time-stepping scheme for the update of Lagrangian flow maps as a Monge-Kantorovich transference problem. This approach provides then a variational principle for the updates of an Eulerian phase Indicatrix in terms of the Wasserstein metric. The Euler-Lagrange equations corresponding to the later variational principle provide the thought after evolution equation for the implicit surfaces.

Suggested Citation

  • Petr Klouček & Michel V. Romerio & Jennifer L. Wightman, 2004. "A Dynamical System Describing Evolution of the Implicit Surfaces in Incompressible Viscous Liquids," Springer Books, in: Miloslav Feistauer & Vít Dolejší & Petr Knobloch & Karel Najzar (ed.), Numerical Mathematics and Advanced Applications, pages 569-579, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18775-9_54
    DOI: 10.1007/978-3-642-18775-9_54
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